Notes

1
Solving Equations Involving Logarithmic and
Exponential Functions
On completion of this module you will be able to

• convert logarithmic with bases other than 10 or
  e
• use the inverse property of exponential and
  logarithmic functions to simplify equations
• understand the properties of logarithms
• use the properties of logarithms to simplify
  equations
• solve exponential and logarithmic equations

2
Bases other than 10 or e

• Most calculators have log x (base 10) and ln x
  (base e).
• How can we solve equations involving bases other
  than 10 or e?
• One way is using the change of base rule

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Example
4
Answer
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Answer
6
Using the inverse property

• When an exponential function and a logarithmic
  function have the same base, they are inverses
  and so effectively cancel each other out.
• Example
• Solve for x

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• We cant divide by log!
• Use the exponential function with the same base
  (10) called taking the anti-log.
• The left and right sides of the equation become
  exponents with a base of 10

8

• Example
• Solve for x
• Answer
• Quick solution is to rearrange using the
  definition of logs
• Alternative
• Now rearrange to isolate the variable

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Take anti-logs
10

• Example
• Solve
• Answer
• We have an exponential function (base e) which we
  can cancel out by taking the logarithm with the
  same base (ln x)

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Properties of logarithms
Example 1 (Since 84 12 ? 7)
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Example 2 Solve for x Answer
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Note that although both 7.0711 and -7.0711
square to give 50, only 7.0711 solves the
original equation. Check as required, but is
undefined. Always check that your answer solves
the original problem!!
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Example
or
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Example
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Example
Note Rules 1 to 4 have been expressed in base 10,
but are equally valid using any base
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This rule also works for any base e.g. since
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• Rule 6 also extends to other bases.
• Whenever we take the log of the same number as
  the base, then the answer is 1.
• e.g.

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• Lets use Rules 3 and 6 to show why Rule 7 is
  true.

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• This uses the concept of log and exponential
  functions as inverses as we discussed earlier.
• This rule also works for other bases.

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• Recall that this is the change of base formula
  used earlier.

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Example
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Example
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Example
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Summary Rules of Logarithms
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Exponential and logarithmic equations
In solving equations which involve exponential
and logarithmic terms, the following properties
allow us to remove such terms and so simplify the
equation.
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Example
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Example
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Example
It doesnt matter whether base 10 or base e is
used, the result will be the same.
Base 10 Base e
The numbers are different but the result is the
same.
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Example
The demand equation for a consumer product
is Solve for p and express your answer in terms
of common logarithms. Evaluate p to two decimal
places when q60.
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Answer
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33
Example
Suppose that the daily output of units of a new
product on the tth day of a production run is
given by Such an equation is called a learning
equation and indicates as time progresses, output
per day will increase. This may be due to a gain
in a workers proficiency at his or her job.
34
Example (continued)
Determine, to the nearest complete unit, the
output on (a) the first day and (b) the tenth day
after the start of a production run. (c) After
how many days will a daily production run of 400
units be reached? Give your answer to the
nearest day.
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Answer

• On the first day or production, t 1, so the
  daily output will be
• When t 10,
• Note that since the answers to parts (a) and (b)
  are the number of units of a new product, we have
  rounded these to the nearest whole unit.

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1. The production run will reach 400 units when
   q 400 or at

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• Notice that the question requires the answer to
  be rounded to the nearest whole day.
• If the answer were round to 8 days,
• so production has not quite reached 400.
• For this reason we round the answer to 9 days,
  even though production will be well passed 400 by
  the end of the 9th day.

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• As always, we must check that the mathematically
  obtained solution answers the original question.